Analytical research of the Hopf bifurcation in the problem of motion of the rattleback

Abstract

We consider the classical problem of the nonholonomic system dynamics — the problem of motion of a heavy rigid body on an perfectly rough horizontal plane. The effect of loss of stability of permanent rotation of a body at a certain critical value of its angular velocity is discussed. It is proved that this effect is accompanied by the occurrence of periodic motions of the body with a frequency close to the critical value, that is, the Hopf bifurcation takes place. It is proved by direct calculation of the first Lyapunov coefficient that the corresponding periodic motions are unstable.